Incenter

An incenter is the center of an incircle, which is a circle tangent to all three sides of a triangle. The trilinear coordinates of this center is 1 : 1 : 1.

Claim: The incenter is located at the concurrent point of the three angle bisectors of a triangle.

Proof: (Draw along if it helps.) For triangle ABC, let X be the intersection of the angle bisector l_A of vertex A, and the angle bisector l_B of vertex B. From point X, construct orthogonal lines to sides AB, BC, and CA, and call the orthogonal intersection points as C', A', and B', respectively. Because of same angles and shared sides, the following triangles form congruent pairs:

AXB' ≅ AXC'
BXA' ≅ BXC'
CXA' ≅ CXB'

As such, A', B', and C' are equidistant from X. Since they are orthogonal intersections, they are the shortest distance from X to the sides. Hence a circle I centered at X containing all points A', B', and C' tangentially on the circumference is the incircle.

Conversely, by examining in reverse the tangent points of any incircle, it becomes clear that the incenter must lie on all angle bisectors, thus proving uniqueness.

A triangle has no one unique center, but the incenter may claim to be the most popular and easy to conceptualize. (The other is the circumcenter.)

What is the largest circle that can fit inside a triangle? There is only one, it turns out to be the incircle. The incircle has a radius r equal to K/s, where K is the area of the triangle, and s is the semiperimeter.

      K      2*K         
r  =  -  = -------                                  (1a) 
      s    (a+b+c)                  

Alternative expressions for the inradius are:

      (a*b*c)
r  = ---------                                      (1b) 
       4*s*R 

        K^2
r  = ---------                                      (1c) 
     rA*rB*rC
 
r  = R*( cos(alpha) + cos(beta) + cos(gamma) -1 )   (1d)
                 

where:

alpha  = triangle's interior angle at vertex A
beta   = interior angle at B
gamma  = interior angle at C
s      = semiperimeter of a triangle, equal to (a+b+c)/2
K      = area of the triangle, equal to sqrt(s*(s-a)*(s-b)*(s-c)) by Heron's Formula
R      = Radius of the circumcircle, also called the circumradius or the exradius
rA     = Radius of the excircle located opposite A
rB     = Radius of the excircle located opposite B
rC     = Radius of the excircle located opposite C

SOURCE: MathWorld, Inradius, MathWorld.

The center of the incircle is called the incenter, denoted as I¹. The incenter's coordinates are:

      (a*Ax + b*Bx + c*Cx)             
Ix = ----------------------       (1a) 
          (a + b + c)                  


      (a*Ay + b*By + c*Cy)             
Iy = ----------------------       (1b) 
          (a + b + c)                  

where:

I = (Ix, Iy)... Cartesian coordinates of the incenter
A = (Ax, Ay)... the coordinates of vertex A of triangle ABC
B = (Bx, By)... the coordinates of vertex B of triangle ABC
C = (Cx, Cy)... the coordinates of vertex C of triangle ABC

Example 1: An acute triangle has vertices A, B, and C at A = (-2,-2), B = (5,3), and C = (1,4). The first things we calculate are the lengths of the sides, a, b, and c, the semiperimeter s, and then the area of the triangle, K.

A = (Ax, Ay) = (-2,-2)
B = (Bx, By) = ( 5, 3)
C = (Cx, Cy) = ( 1, 4)

a = length of the side opposite vertex A
  = length of the side BC
  = sqrt((Cx-Bx)^2 + (Cy-By)^2)
  = sqrt((1 - 5)^2 + (4 - 3)^2)
  = sqrt(16 + 1) = sqrt(17)
  = 4.1
b = length of the side opposite vertex B
  = length of the side CA
  = sqrt ((Ax-Cx)^2 + (Ay-Cy)^2)
  = sqrt ((-2 - 1)^2 + (-2 - 4)^2)
  = sqrt(9 + 36) = sqrt(45)
  = 6.7
c = length of the side opposite vertex C
  = length of the side AB
  = sqrt ((Bx-Ax)^2 + (By-Ay)^2)
  = sqrt ((5 - -2)^2 + (3 - -2)^2)
  = sqrt(49 + 25) = sqrt(74)
  = 8.6
s = (a+b+c)/2
  = (4.1 + 6.7 + 8.6)/2
  = 9.7
K = sqrt(s*(s-a)*(s-b)*(s-c))
  = sqrt(9.7*(9.7-4.1)*(9.7-6.7)*(9.7-8.6))
  = 13.5

Now we can calculate the incenter's coordinates.

      (4.1*(-2) + 6.7*5 + 8.6*1)    33.9      
Ix = ---------------------------- = ---- = 1.7
          (4.1 + 6.7 + 8.6)         19.4      

      (4.1*(-2) + 6.7*3 + 8.6*4)    46.3      
Iy = ---------------------------- = ---- = 2.4
          (4.1 + 6.7 + 8.6)         19.4      

I = (Ix, Iy) = (1.7, 2.4)

Point of Concurrency of Angle Bisectors: The incenter is the point of concurrency of the angle bisectors. If you bisect every angle, and you draw the ray from the vertex to the opposite side, then, amazingly, every one of these line segments intersects at one point!

Now you would suspect that two of these lines would intersect at a point, but when the third line segment intersects at exactly the same point, you know something's up.

Euclid proved that the angle bisectors all meet at the same point, always, for every possible triangle (which is the definition for the point of concurrency). He did this in Book 4, Proposition 4 of his magnum opus, The Elements.

There's a wonderful Web site devoted to Euclid's Elements here, and the wonderfulness is enhanced by its illustrations, which turn out to be Java applets. (You must permit a Java applet to run on your browser.) You can move the vertices of a triangle around and see how the incircle moves with the moving vertices. The incircle is always the intersection of the angle bisectors.

(I can see this a million times and never get bored. The sheer audacity of mathematics to have proved that this is always the case!)

Touch Points: The incircle touches the triangle at three contact points, commonly called touch points. Let's label them T_A, T_B, and T_C.

If you draw a line segment connecting each touch point to the incenter, each line segment will make a right angle to the side it touches. (It has to - the radius of a circle is always perpendicular to the tangent line at the point of contact.)

The line segments are all radii of the incircle, so their lengths are all equal to r.

The distances of the touch points to the vertices is quite interesting. The distance from vertex A is the same to the two closest touchpoints. Let's label this quantity x. Similarly, let's label distances y as the distance between the two closest touchpoints to B, and z to C. It turns out that the distances are as follows:

   
x = s - a                         (2a) 
y = s - b                         (2b) 
z = s - c                         (2c) 

which means that now we can compute the point coordinates:

T_Ax = Bx + (y/a)*(Cx - Bx)       (3a)
T_Ay = By + (y/a)*(Cy - By)       (3a)
T_Bx = Cx + (z/b)*(Ax - Cx)       (3c) 
T_By = Cy + (z/b)*(Ay - Cy)       (3d) 
T_Cx = Ax + (x/c)*(Bx - Ax)       (3e) 
T_Cy = Ay + (x/c)*(By - Ay)       (3f) 

This is nicely illustrated and explained by Alexander Bogomolny here.

Example 2: Using the same acute as in Example 1, find the contact points. First, we compute the distances x, y, and z, then we compute the touchpoint coordinates.

A = (Ax, Ay) = (-2,-2)
B = (Bx, By) = ( 5, 3)
C = (Cx, Cy) = ( 1, 4)
a = 4.1
b = 6.7
c = 8.6
s = 9.7

x = 9.7 - 4.1 = 5.6
y = 9.7 - 6.7 = 3.0
z = 9.7 - 8.6 = 1.1

T_Ax =  5 + (3.0/4.1)*(1-5)    = 2.1
T_Ay =  3 +     0.73 *(4-3)    = 3.7
T_Bx =  1 + (1.1/6.7)*(-2 - 1) = 0.5
T_By =  4 +     0.17 *(-2 - 4) = 3.0
T_Cx = -2 + (5.6/8.6)*(5 - -2) = 2.6
T_Cy = -2 +     0.65 *(3 - -2) = 1.3

TA = (T_Ax, T_Ay) = (2.1, 3.7)
TB = (T_Bx, T_By) = (0.5, 3.0)
TC = (T_Cx, T_Cy) = (2.6, 1.3)

I created a little spreadsheet in Excel that would graph the triangle, the incircle, the incenter, and the touch points just to verify that these formulas yielded the right values. A picture of this is on my homenode. If I happen to change the homenode picture and it's not there, I'll be glad to email you the jpeg image. Just leave me a message here and include your email address.

Barycentric Coordinates: The barycentric coordinates of I are a:b:c.

Trilinear Coordinates: The trilinear coordinates of I are 1:1:1.

NOTES

1. Bold indicates that these are points.

Everything2 Writeups: Articles on (topic)

1. tongpoo, incenter, Feb. 7, 2002
2. tongpoo, incircle, Sep. 19, 2002
3. tongpoo, triangle, Feb. 8, 2002

References: Useful books and references on geometry

1. H.S.M. Coxeter, Introduction to Geometry, 2nd Ed., (c) 1969
   *SIGH* What a magnificent book.
   § 1.4, “The Medians and the Centroid,” p. 10
   § 1.5, “The Incircle and the Circumcircle,” pp. 11-16
   § 1.6, “The Euler Line and the Orthocenter,” p. 17
   § 1.7, “The Nine Point Circle,” pp. 18-20
   § 1.9, “Morley’s Theorem,” pp. 23-25
   § 1.6, “The Euler Line and the Orthocenter,” p. 17
2. Dan Pedoe, Geometry: A Comprehensive Course
3. C. Stanley Ogilvy, Excursions in Geometry, (c) 1969
   An elegant, thin discourse on geometry. Surprises in every chapter, even for the mathematically astute geometer.
   Ch. 8, Some Euclidean Topics,
   § 8.1, “A navigation Problem,” p. 111
   § 8.2, “A three-circle Problem,” p. 115
   § 8.3, “The Euler line,” p. 117
   § 8.4, “The nine-point circle,” p. 119
   § 8.5, “A triangle problem,” p. 120
4. J.L. Heilbron, Geometry Civilized, ©2000
5. Robert Dixon, Mathographics, ©1987
6. David Wells, Ed., The Penguin Dictionary of Curious and Interesting Geometry, ©1991
7. Hans Schwerdtfeger, Geometry of Complex Numbers, ©1962
8. Roland Deaux, Introduction to the Geometry of Complex Numbers, Howard Eves, Tr., Originally published in 1956
   § 38, “Centroid of a triangle,” p. 60
   § 39, “Algebraic value for the area of a triangle,” p. 60
9. Bruce Meserve, Fundamental Concepts of Geometry, ©1983
10. Will Dunham, Journey Through Genius, ©1990
    Ch 5, “Heron’s formula for triangular area,” p. 113
11. Melvin Hausner, A Vector Space Approach to Geometry, ©1965
12. M. de Berg, M. van Kreveld, M. Overmans, and O. Schwartzkopf, Computational Geometry, 2nd Ed, ©2000
13. Gerald Farin and Dianne Hansford, The Geometry Toolbox, ©1998
    Ch. 3, 2D Lines
    § 3.6, “Distance of a point to a line,” p. 40
    § 3.7, “The foot of a point,” p. 44
    § 3.8, “Computing intersections,” p. 45
    Ch. 8, Breaking it up: Triangles
    § 8.1, “Barycentric coordinates,” p. 126
    § 8.2, “Affine invariance,” p. 128
    § 8.3, “Some special points,” p. 128
14. Clifford Pickover, The Math Book, ©2009
    c.600 BC, “Pythagorean Theorem and Triangles,” p. 40
    1639, “Projective Geometry,” p. 142
    1899, “Pick’s Theorem,” p. 294
    1899, “Morley’s Trisector Theorem,” p. 296
    
15. Keith Devlin, The Language of Mathematics, ©2000
16. Daniel Zwillinger, Ed., The CRC Standard Mathematical Tables and Formulae, 30th Ed, ©1996
    Ch. 4, Geometry,
    § 4.5.1, “Triangles,” p. 271
    § 4.6, “Circles,” p. 277
    
17. Siobhan Roberts, King of Infinite Space: Donald Coxeter, the Man Who Saved Geometry, ©2006
18. Alfred S. Posamentier and Charles T. Salkind, Challenging Problems in Geometry, ©1988
19. Edna E. Kramer, The Nature and Growth of Modern Mathematics, ©1970, 1981
    Figure 2.7, the square law spiral, p. 28
20. Hans Rademacher and Otto Toeplitz, The Enjoyment of Mathematics, Published in 1957 by the Princeton University Press
    § 5, “A minimum property of the pedal triangle,” p. 27 (minimum path length of light when crossing a boundary)
    § 14, “Pythagorean numbers and Fermat’s Theorem,” p. 88
    § 16, “The spanning circle of a finite set of points,” p. 108 (H.W.E. Jung’s theorem)
    § 26, “A characteristic property of the circle,” p. 160
    § 28, “The indispensability of the compass for the constructions of elementary geometry,” p. 177

Internet References

1. David E. Joyce, Euclid's Elements, Book IV, Prop. 4. Euclid's proof and Java applets showing the incenter of a triangle being the point of concurrency of the three angle bisectors.
2. David E. Joyce, Homepage at Clark University. Joyce is a professor of Mathematics and Computer Science at Clark University, Worcester, MA. He has rendered Euclid's Elements for the computer and added numerous Java applets in the next reference. In my opinion, his site is one of the greatest reasons that geometers should use the Internet.
3. David E. Joyce, Euclid's Elements. Euclid's Elements on steroids. The java applets make geometry and geometric proofs come alive. I defy you to read a few of his propositions, try out the applets, and then not be amazed at Euclid's genius.
4. P. Ballew, “Orthocenter of a triangle”
5. Wikipedia, "Triangle"
6. Wikipedia, "Incircle and Excircles of a Triangle"
7. Weisstein, Eric W. "Incenter" From MathWorld--A Wolfram Web Resource.
8. Weisstein, Eric W. "Inradius" From MathWorld--A Wolfram Web Resource.
9. Weisstein, Eric W. "Triangle" From MathWorld--A Wolfram Web Resource.
10. Alexander Bogomolny, "Incircle and Excircles of a Triangle" From Cut The Knot--mathematical topics. Cut the Knot has a full range of geometric topics. It is a reference I use quite frequently, as the language is clean and to the point, without being unnecessarily over-mathematical, as Mathworld's information tends to be.

In*cen"ter (?), n. Geom.

The center of the circle inscribed in a triangle.

 

© Webster 1913.